In the special case of a Gaussian likelihood \(\pi _{Y\vert X}\)  and linear
\(\widehat{S}\) , the estimator
\(\pi _{\widehat{S}}\)  of the posterior distribution
becomes
Gaussian.
The corresponding
KR rearrangement that pushes
forward \(\mathcal {N}({\bf 0}, {\bf I}_n)\)  to \(\pi _{\widehat{S}}\)  is linear, meaning that without loss of generality
we can solve
(REF ) over the space
of linear maps. In this case, (REF ) is
a convex (quadratic) problem with a closed
form expression for \(\smash{\widehat{\mathcal {T}}}\) ,
and the deterministic map filter reduces
to a particular
ensemble square-root filter [1]}.
